Two men on either side of a pole of 30 m high observe its top at angles of elevation α and β respectively. The distance between the two men is \(40\sqrt{3}\) m and the distance between the first man at A and the pole is \(30\sqrt{3}\) m.
Based on the above information, answer the question:
tan(α + β) =
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Infinite (does not exist)
The pole is BD = 30 m, with AD = \(30\sqrt{3}\) m and \(DC = 40\sqrt{3} - 30\sqrt{3} = 10\sqrt{3}\) m.
\( \displaystyle \tan\alpha = \frac{30}{30\sqrt{3}} = \frac{1}{\sqrt{3}},\quad \alpha = \frac{\pi}{6} \)
\( \displaystyle \tan\beta = \frac{30}{10\sqrt{3}} = \sqrt{3},\quad \beta = \frac{\pi}{3} \)
So
\( \displaystyle \alpha + \beta = \frac{\pi}{2} \)
and \(\tan\frac{\pi}{2}\) is not defined. In the formula \(\tan(\alpha+\beta) = \frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\), the denominator is \(1 - \frac{1}{\sqrt{3}}\times\sqrt{3} = 0\).